Abstract
If one compares the literature on abstract Harmonic Analysis or publications with the word “Fourier Analysis” in the title with books describing the foundations of digital signal processing or systems theory one may easily get the impression that they describe two different worlds having little in common except for some vocabulary, indicating their joint roots.
It is the purpose of this article to shed some light on the existing and sometimes buried interconnections between these two branches of Fourier Analysis. We suggest to take a broader perspective on Harmonic Analysis, re-emphasizing how they are tied together in many ways and how numerical experiments may help to understand concepts of Harmonic Analysis. Based on our experience we firmly believe that Fourier Analysis provides opportunities to relate relevant mathematical theorems to valuable algorithms, which also have to be properly implemented (typically making use of the FFT, the Fast Fourier Transform). Thus this note tries to bridge the gap between pure mathematics and real-world engineering applications.
In a previous paper (H. G. Feichtinger, Elements of Postmodern Harmonic Analysis, pages 1 – 27, Springer, 2015) the author has already outlined some ideas in this direction, by introducing the idea of Conceptual Harmonic Analysis, as a link between the two worlds. The best way to describe the connections between the two worlds is by means of generalized functions (often called distributions). We will indicate that the Banach Gelfand triple \( (\boldsymbol{S}_{\!0},\boldsymbol{L}^{2},\boldsymbol{S}_{\!0}\!')(\mathbb{R}^{d}) \), which is based on a particular Banach algebra of continuous functions, namely the Segal algebra \( \big(\boldsymbol{S}_{\!0}(\mathbb{R}^{d}),\|\mbox{ $\,\cdot \,$}\|_{\boldsymbol{S}_{\!0}}\big) \), provides convenient vehicle to express many otherwise “soft transition” in a distributional sense. In the context of the space \( \boldsymbol{S}_{\!0}(\mathbb{R}^{d}) \) of test functions approximation by finite sequences makes perfect sense and one can justify many transitions between the two worlds (continuous versus finite signals) in a clear mathematical context (see H. G. Feichtinger and N. Kaiblinger, Quasi-interpolation in the Fourier algebra, J. Approx. Theory, 144(1):103–118, 2007).The current text also contains various considerations of a general nature, which are supposed to stimulate discussions and reflection of the work done in the field of abstract and applied Harmonic Analysis in general.
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References
I. Amidror. Mastering the Discrete Fourier Transform in one, two or several Dimensions. Pitfalls and Artifacts. London: Springer, 2013.
P. Antosik, J. Mikusinski, and R. Sikorski. Theory of Distributions. The Sequential Approach.. Elsevier Scientific Publishing Company, 1973.
J. J. Benedetto and A. I. Zayed. Sampling, Wavelets, and Tomography. Birkhäuser, 2004.
R. N. Bracewell. The Fourier Transform and its Applications. McGraw-Hill Book Company, Auckland etc., 2nd ed., 3rd printing. International Student Edition. edition, 1983.
W. L. Briggs and V. E. Henson. The DFT. An Owner’s Manual for the Discrete Fourier Transform. SIAM, Philadelphia, PA, 1995.
P. Butzer, P. Ferreira, J. Higgins, G. Schmeisser, and R. L. Stens. The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals – their interconnections. Appl. Anal., 90(3–4):431–461, 2011.
P. Butzer, P. Ferreira, G. Schmeisser, and R. L. Stens. The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis. Result. Math., 59(3–4):359–400, 2011.
P. L. Butzer and A. Gessinger. The approximate sampling theorem, Poisson’s sum formula, a decomposition theorem for Parseval’s equation and their interconnections. Ann. Numer. Math., 4(1–4):143–160, 1997.
P. L. Butzer and R. J. Nessel. Fourier Analysis and Approximation. Vol. 1: One-dimensional Theory. Birkhäuser, Stuttgart, 1971.
P. L. Butzer and R. L. Stens. The Poisson summation formula, Whittaker’s cardinal series and approximate integration. In Approximation Theory, 2nd Conf, Edmonton/Alberta 1982, CMS Conf Proc, Volume 3, pages 19–36, 1983.
G. Cariolaro. Unified Signal Theory. Springer, London, 2011.
J. Cooley and J. Tukey. An algorithm for the machine calculation of complex Fourier series. Math. Comp., 19:297–301, 1965.
E. Cordero, H. G. Feichtinger, and F. Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-differential Operators, Volume 1949 of Lecture Notes in Mathematics, pages 1–33. Springer, Berlin, 2008.
A. Deitmar. A First Course in Harmonic Analysis. Universitext. Springer, New York, NY, 2002.
A. Deitmar and S. Echterhoff. Principles of Harmonic Analysis. New York: Springer, 2009.
H. G. Feichtinger. Choosing Function Spaces in Harmonic Analysis, Volume 4 of Excursions in Harmonic Analysis. The February Fourier Talks at the Norbert Wiener Center. Birkhäuser, 2015.
H. G. Feichtinger. Elements of Postmodern Harmonic Analysis, pages 1 – 27. Springer, 2015.
H. G. Feichtinger and K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal., 86(2):307–340, 1989.
H. G. Feichtinger and K. Gröchenig. Multidimensional irregular sampling of band-limited functions in L p-spaces. Conf. Oberwolfach Feb. 1989, pages 135–142, 1989.
H. G. Feichtinger and K. Gröchenig. Iterative reconstruction of multivariate band-limited functions from irregular sampling values. SIAM J. Math. Anal., 23(1):244–261, 1992.
H. G. Feichtinger and N. Kaiblinger. Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc., 356(5):2001–2023, 2004.
H. G. Feichtinger and N. Kaiblinger. Quasi-interpolation in the Fourier algebra. J. Approx. Theory, 144(1):103–118, 2007.
H. G. Feichtinger and W. Kozek. Quantization of TF lattice-invariant operators on elementary LCA groups. In H. G. Feichtinger and T. Strohmer, editors, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., pages 233–266. Birkhäuser Boston, Boston, MA, 1998.
H. G. Feichtinger and F. Luef. Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis. Collect. Math., 57(Extra Volume (2006)):233–253, 2006.
H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 123–170, 1998, Birkhäuser Boston.
J. V. Fischer. On the duality of discrete and periodic functions. Mathematics, 3(2):299–318, 2015.
G. B. Folland. Harmonic Analysis in Phase Space. Princeton University Press, Princeton, N.J., 1989.
G. B. Folland. A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. viii, Boca Raton, FL, 1995.
M. Frigo and S. Johnson. FFTW: An adaptive software architecture for the FFT. In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, 1998., Volume 3, pages 1381–1384, 1998.
M. Frigo and S. Johnson. FFTW User’s Manual. Massachusetts Institute of Technology, 1998.
M. Frigo and S. Johnson. The design and implementation of FFTW 3. Proceedings of the IEEE, 93(2):216–231, 2005.
D. Gabor. Theory of communication. J. IEE, 93(26):429–457, 1946.
C. Gasquet and P. Witomski. Fourier Analysis and Applications. Filtering, Numerical Computation, Wavelets. Transl. from the French by R. Ryan. Springer, 1999.
L. Grafakos. Classical Fourier Analysis (Second Edition), Volume 249 of Graduate Texts in Mathematics. Springer, New York, 2008.
L. Grafakos. Modern Fourier Analysis(Second Edition), Volume 250 of Graduate Texts in Mathematics. Springer, New York, 2009.
K. Gröchenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston, MA, 2001.
K. Gröchenig and M. Leinert. Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc., 17:1–18, 2004.
E. Hewitt and K. A. Ross. Abstract Harmonic Analysis. Vol. 1: Structure of Topological Groups; Integration Theory; Group Representations. 2nd ed. Springer-Verlag, Berlin-Heidelberg-New York, 1979.
E. Hewitt and K. A. Ross. Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Springer, Berlin, Heidelberg, New York, 1970.
J.-P. Kahane and P.-G. Lemarie Rieusset. Remarks on the Poisson summation formula (Remarques sur la formule sommatoire de Poisson). Studia Math., 109:303–316, 1994.
N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11(1):25–42, 2005.
D. W. Kammler. A First Course in Fourier Analysis. Prentice Hall, 1999.
R. P. Kanwal. Generalized Functions. Theory and Applications. 3rd Revised ed. Birkhäuser, 2004.
Y. Katznelson. An Introduction to Harmonic Analysis. 3rd Corr. ed. Cambridge University Press, 2004.
H. L. Lebesgue. Lecons sur l’intégration et la Recherche des Fonctions Primitives. Cambridge Library Collection. Cambridge University Press, Cambridge, 2009.
L. Loomis. An Introduction to Abstract Harmonic Analysis. Van Nostrand and Co., 1953.
B. Luong. Fourier Analysis on Finite Abelian Groups. Birkhäuser, 2009.
J. Ramanathan. Methods of Applied Fourier Analysis. Birkhäuser, Boston, 1998.
K. R. Rao and P. Yip. Discrete Cosine Transform. Algorithms, Advantages, Applications. Academic Press., Boston, MA, 1990.
H. Reiter. Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford, 1968.
H. Reiter and J. D. Stegeman. Classical Harmonic Analysis and Locally Compact Groups. 2nd ed. Clarendon Press, Oxford, 2000.
J. Romulad. Fourier approach to digital holography. Volume 4607, pages 161–167. SPIE, 2002.
W. Rudin. Fourier Analysis on Groups. Interscience Publishers, New York, London, 1962.
I. Sandberg. The superposition scandal. Circuits Syst. Signal Process., 17(6):733–735, 1998.
I. Sandberg. Causality and the impulse response scandal. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50(6):810–813, Jun 2003.
I. Sandberg. Continuous multidimensional systems and the impulse response scandal. Multidimensional Syst. Signal Process., 15(3):295–299, 2004.
R. Shakarchi and E. M. Stein. Fourier Analysis: An Introduction. Princeton Lectures in Analysis. Princeton University Press, 2003.
E. Stade. Fourier analysis. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, 2005.
A. Terras. Fourier Analysis on Finite Groups and Applications., Volume 43 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.
R. M. Trigub and E. S. Belinsky. Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, 2004.
R. Vallee. The ‘epsilon distribution’or the antithesis of Dirac’s delta. Cybernetics and Systems Research, 92:97–102, 1992.
R. Vallee. About Wiener’s generalized Harmonic Analysis. Kybernetes, 23(6/7):65–70, 1994.
R. Vallee. Generalized Harmonic Analysis and epsilon-distribution. Cybernetica, 37(3–4): 377–385, 1994.
R. Vallee. On certain distributions met in signal theory and other domains of systems science. Systems Science, 30(2):5–10, 2004.
D. Voelz. Computational Fourier Optics. A MATLAB Tutorial. Tutorial text. Vol. 89. SPIE, 2011.
H. J. Weaver. Applications of Discrete and Continuous Fourier Analysis. Wiley-Interscience, 1983.
H. J. Weaver. Theory of Discrete and Continuous Fourier Analysis. Wiley-Interscience, 1989.
A. Weil. L’integration dans les Groupes Topologiques et ses Applications. Hermann and Cie, Paris, 1940.
M. Wong. Discrete Fourier analysis. Pseudo-Differential Operators. Theory and Applications 5. Basel: Birkhäuser. viii, 176 p., 2011.
A. I. Zayed. Advances in Shannon‘s Sampling Theory. CRC Press, Boca Raton, FL, 1993.
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Feichtinger, H.G. (2016). Thoughts on Numerical and Conceptual Harmonic Analysis. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27873-5_9
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